NDLI: Linear Quadratic Differential Games: Closed Loop Saddle Points

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Sbarba, Olivier Dello Delfour, Michel C.
Copyright Year	2009
Abstract	The object of this paper is to revisit the results of Bernhard [J. Optim. Theory Appl., 27 (1979), pp. 5169] on two-person zero-sum linear quadratic differential games and generalize them to utility functions without positivity assumptions on the matrices acting on the state variable and to linear dynamics with bounded measurable data matrices. Our paper specializes to state feedback via Lebesgue measurable affine closed loop strategies with possible non-$L^2$-integrable singularities. After sharpening the recent results of Delfour [SIAM J. Control Optim., 46 (2007), pp. 750774] on the characterization of the open loop lower and upper values of the game, it first deals with $L^2$-integrable closed loop strategies and then with the larger family of strategies that may have non-$L^2$-integrable singularities. A new conceptually meaningful and mathematically precise definition of a closed loop saddle point is introduced to simultaneously handle state feedbacks of the $L^2$ type and smooth locally bounded ones, except at most in the neighborhood of finitely many instants of time. A necessary and sufficient condition is that the free end problem be normalizable almost everywhere. This relaxation of the classical notion allows singularities in the feedback law at an infinite number of instants, including accumulation points that are not isolated. A complete classification of closed loop saddle points is given in terms of the convexity/concavity properties of the utility function, and connections are given with the open loop lower value, upper value, and value of the game.
Starting Page	3138
Ending Page	3166
Page Count	29
File Format	PDF
ISSN	03630129
DOI	10.1137/070696593
e-ISSN	10957138
Issue Number	6
Volume Number	47
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-01-07
Access Restriction	Subscribed
Subject Keyword	linear quadratic differential game 2-person games open loop and closed loop strategies integrable singularities Differential games saddle point Dynamic games zero sum value of a game Riccati differential equation two person
Content Type	Text
Resource Type	Article
Subject	Control and Optimization Applied Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in